Hi, do you mean, when the visitor no take any action in two minutes in the page results, exists an refresh on the results, right? Is not bad idea.
You can do it with html code, in the <head> tag, in your search.php file:
<head>
<meta http-equiv="Refresh" content="120; url=" />
</head>
content="is the number of seconds you need for auto refresh the results."
Try and see if works for what you need
Greetings